Yakubovich, Fradkov, Hill and Proskurnikov have used the Yaku-bovich Frequency Theorem to prove that a strictly dissipative linear-quadratic control process with periodic coefficients admits a storage function, and various related results. We extend their analysis to the case when the coefficients are bounded uniformly continuous functions.

Under the assumption of lack of uniform controllability for a family of time-dependent linear control systems, we study the dimension, topological structure and other dynamical properties of the sets of null controllable points and of the sets of reachable points. In particular, when the space of null controllable vectors has constant dimension for all the systems of the family, we find a closed invariant subbundle where the uniform null controllability holds. Finally, we associate a family of linear Hamiltonian systems to the control family and assume that it has an exponential dichotomy in order to relate the space of null controllable vectors to one of the Lagrange planes of the continuous hyperbolic splitting.

Generally speaking, the term nonautonomous dynamics refers to the systematic use of dynamical tools to study the solutions of differential or difference equations with time-varying coefficients. The nature of the time variance may range from periodicity at one extreme, through Bohr almost periodicity, Birkhoff recurrence, Poisson recurrence etc. to stochasticity at the other extreme. The ``dynamical tools'' include almost everywhere Lyapunov exponents, exponential splittings, rotation numbers, and the theory of cocycles, but are by no means limited to these. Of course in practise one uses whatever ``works'' in the context of a given problem, so one usually finds dynamical methods used in conjunction with those of numerical analysis, spectral theory, the calculus of variations, and many other fields. The reader will find illustrations of this fact in all the papers of the present volume.

A detailed dynamical study of the skew-product semiflows induced by
families of AFDEs with infinite delay on a Banach space is
carried over. Applications are given for families of non-autonomous
quasimonotone reaction-diffusion PFDEs with delay in the nonlinear
reaction terms, both with finite and infinite delay. In this
monotone setting, relations among the classical concepts of sub and
super solutions and the dynamical concept of semi-equilibria are
established, and some results on the existence of minimal semiflows
with a particular dynamical structure are derived.

We consider the skew-product semiflow induced by a family of
finite-delay functional differential equations and we characterize
the exponential stability of its minimal subsets. In the case of
non-autonomous systems modelling delayed cellular neural networks,
the existence of a global exponentially attracting solution is
deduced from the uniform asymptotical stability of the null
solution of an associated non-autonomous linear system.